Optimal. Leaf size=43 \[ -\frac {a^2 \coth ^3(c+d x)}{3 d}-\frac {a (a+2 b) \coth (c+d x)}{d}+x (a+b)^2 \]
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Rubi [A] time = 0.07, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3670, 461, 207} \[ -\frac {a^2 \coth ^3(c+d x)}{3 d}-\frac {a (a+2 b) \coth (c+d x)}{d}+x (a+b)^2 \]
Antiderivative was successfully verified.
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Rule 207
Rule 461
Rule 3670
Rubi steps
\begin {align*} \int \coth ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^2}{x^4 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2}{x^4}+\frac {a (a+2 b)}{x^2}-\frac {(a+b)^2}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a (a+2 b) \coth (c+d x)}{d}-\frac {a^2 \coth ^3(c+d x)}{3 d}-\frac {(a+b)^2 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=(a+b)^2 x-\frac {a (a+2 b) \coth (c+d x)}{d}-\frac {a^2 \coth ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.62, size = 65, normalized size = 1.51 \[ -\frac {\coth (c+d x) \left (a \left (a \coth ^2(c+d x)+3 a+6 b\right )-3 (a+b)^2 \tanh ^{-1}\left (\sqrt {\tanh ^2(c+d x)}\right ) \sqrt {\tanh ^2(c+d x)}\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 197, normalized size = 4.58 \[ -\frac {2 \, {\left (2 \, a^{2} + 3 \, a b\right )} \cosh \left (d x + c\right )^{3} + 6 \, {\left (2 \, a^{2} + 3 \, a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 4 \, a^{2} + 6 \, a b\right )} \sinh \left (d x + c\right )^{3} - 6 \, a b \cosh \left (d x + c\right ) + 3 \, {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x - {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 4 \, a^{2} + 6 \, a b\right )} \cosh \left (d x + c\right )^{2} + 4 \, a^{2} + 6 \, a b\right )} \sinh \left (d x + c\right )}{3 \, {\left (d \sinh \left (d x + c\right )^{3} + 3 \, {\left (d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 103, normalized size = 2.40 \[ \frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (d x + c\right )} - \frac {4 \, {\left (3 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 6 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} + 3 \, a b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 59, normalized size = 1.37 \[ \frac {a^{2} \left (d x +c -\coth \left (d x +c \right )-\frac {\left (\coth ^{3}\left (d x +c \right )\right )}{3}\right )+2 a b \left (d x +c -\coth \left (d x +c \right )\right )+b^{2} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 114, normalized size = 2.65 \[ \frac {1}{3} \, a^{2} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + 2 \, a b {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + b^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 175, normalized size = 4.07 \[ x\,{\left (a+b\right )}^2-\frac {\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2+b\,a\right )}{3\,d}-\frac {4\,a\,b}{3\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\frac {4\,\left (a^2+b\,a\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^2+b\,a\right )}{3\,d}-\frac {8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}-\frac {4\,\left (a^2+b\,a\right )}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2} \coth ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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